# Equivalence of Definitions of Strict Ordering on Integers

## Theorem

The following definitions of the concept of **Strict Ordering on Integers** are equivalent:

### Definition 1

The integers are strictly ordered on the relation $<$ as follows:

- $\forall x, y \in \Z: x < y \iff y - x \in \Z_{>0}$

That is, **$x$ is less than $y$** if and only if $y - x$ is (strictly) positive.

### Definition 2

The integers are strictly ordered on the relation $<$ as follows:

Let $x$ and $y$ be defined as from the formal definition of integers:

- $x = \eqclass {x_1, x_2} {}$ and $y = \eqclass {y_1, y_2} {}$ where $x_1, x_2, y_1, y_2 \in \N$.

Then:

- $x < y \iff x_1 + y_2 < x_2 + y_1$

where:

- $+$ denotes natural number addition
- $a < b$ denotes natural number ordering $a \le b$ such that $a \ne b$.

## Proof

Let $x, y \in \Z$ such that $x < y$.

Let $x$ and $y$ be defined as from the formal definition of integers:

- $x = \eqclass {x_1, x_2} {}$ and $y = \eqclass {y_1, y_2} {}$ where $x_1, x_2, y_1, y_2 \in \N$.

### $(1)$ implies $(2)$

Let $<$ be a strict ordering on integers by definition $1$.

Then by definition:

- $y - x$ is non-negative

That is:

- $\map \PP {y - x}$

where $\PP$ is the strict positivity property.

Thus:

- $\eqclass {y_1, y_2} {} - \eqclass {x_1, x_2} {}$ is strictly positive

By definition of integer subtraction:

- $\eqclass {y_1, y_2} {} + \eqclass {x_2, x_1} {}$ is strictly positive

and by the formal definition of integers:

- $\eqclass {y_1 + x_2, y_2 + x_1} {}$ is strictly positive

We have that $y_1 + x_2$ and $y_2 + x_1$ are natural numbers.

Thus by definition of natural number ordering:

- $y_1 + x_2 > y_2 + x_1$

Thus $<$ is a strict ordering on integers by definition $2$.

$\Box$

### $(2)$ implies $(1)$

Let $<$ be a strict ordering on integers by definition $2$.

Then by definition:

- $x_1 + y_2 < x_2 + y_1$

That is:

- $x_2 + y_1 > x_1 + y_2$

Hence by the formal definition of integers:

- $\eqclass {y_1 + x_2, y_2 + x_1} {}$ is strictly positive

By definition of integer addition:

- $\eqclass {y_1, y_2} {} + \eqclass {x_2, x_1} {}$ is strictly positive

By definition of integer subtraction:

- $\eqclass {y_1, y_2} {} - \eqclass {x_1, x_2} {}$ is strictly positive

That is:

- $\map \PP {y - x}$

where $\PP$ is the strict positivity property.

Thus $\le$ is a strict ordering on integers by definition $1$.

$\blacksquare$